Re: Understanding a solution -- intro to ring theory -- ideals

The first paragraph is just saying the ideals generated by the units in the ring is the whole ring correct?

Right.

Also, the principal ideals generated by 2, 4 and 8 are all the same correct? So those are three ideals or just one?
So, in the final answer there are 4 ideals in total right?

They are all the same ideal. There are four ideals. In general (this is the fourth isomorphism theorem) given an ideal of a ring then the set of ideals is the set . In particular, in a (or more generally a PID) this says that the ideals of are of the form where , or equivalently . From this it's easy to deduce that the number of ideals of is the number of (positive) divisors of . So, as a special case .

And finally why do we care about Ideals so much? What use do they even have besides giving students of average intelligence a giant headache in intro abstract algebra course lol?

Ideals encompass much of the theory of rings for multiple reasons. In one direction (the commutative algebraic direction) the ideals of a ring can function as "ideal" (get it now) elements of the ring. So, while a ring may not have unique factorization there may be a theory of unique factorization for its ideals.

In a different (more categorical) direction, we care mostly about rings up to isomorphism. In fact, what rings can be homomorphic images of a given ring, tell us a lot about the ring. Thus, we would like to know what rings have the property that is a homomorphic image of our ring. But, the first isomorphism gives us the answer to this question--the homomorphic images of a ring are precisely the quotient rings of our ring. It's clear then that quotient rings are of a huge importance in ring theory, and since one can define ideals as being precisely the (non-unital) subrings of a given ring for which quotient rings "make sense" ideals become equally important.